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求一算術

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Text ID
KR3fc041
Date
1800–1810
Dynasty
清

Qiúyī suànshù 求一算術

Mathematical Method of the Seeking-One Procedure by 張敦仁 (撰)

About the work

張敦仁 Zhāng Dūnrén’s (1754–1834) systematic Qián-Jiā exposition of 秦九韶 Qín Jiǔsháo’s Dàyǎn qiúyī shù 大衍求一術 (the Chinese Remainder Theorem), in 4 juàn. The first thorough Qīng exposition of this fundamental Sòng-Yuán algorithm, preceding 焦循 Jiāo Xún’s KR3fc044 Tiānyuányī shì and forming part of the broader Yáng-zhōu-circle project of recovering the Sòng-Yuán mathematical methods.

Abstract

The Dàyǎn qiúyī shù — Qín Jiǔsháo’s 1247 systematic algorithm for solving systems of linear congruences (now known as the Chinese Remainder Theorem) — is one of the great pre-modern Chinese mathematical achievements, but its presentation in the KR3fc008 Shùshū jiǔzhāng / KR3f0041 Shùxué jiǔzhāng is highly compressed. Qín Jiǔsháo had given the general algorithm together with worked examples, but had not provided a full conceptual exposition of why the algorithm works; by the late Míng the procedural details had become inaccessible to working mathematicians.

Zhāng Dūnrén’s task in the Qiúyī suànshù was the recovery of the substantive algorithm. The four juàn cover:

(1) The basic problem of linear congruences and the special case of pair-wise coprime moduli (where the algorithm is straightforward).

(2) The general case where the moduli share common factors — the harder case that Qín Jiǔsháo had systematically addressed but whose treatment had become obscure.

(3) The qiúyī (seeking-one) sub-procedure — the procedure for finding, for a given pair of coprime integers (a, b), the smallest positive integer x such that ax ≡ 1 (mod b). This is the conceptual heart of the algorithm; Zhāng Dūnrén supplies a full step-by-step exposition with worked examples.

(4) Applied problems: Qín Jiǔsháo’s original 81 problems retraversed with the recovered algorithmic clarity, plus additional Zhāng-supplied problems illustrating various edge cases and applications.

The work was the first systematic Qīng exposition of the qiúyī method and made the algorithm newly accessible to mid-Qīng working mathematicians. It is one of the principal documents of the Jiā-Dào-era Sòng-Yuán mathematical recovery — the project, coordinated across the Yángzhōu mathematical circle, of restoring the lost algebraic methods of the Sòng-Yuán tradition to active circulation. Zhāng Dūnrén’s qiúyī exposition is the foundational text on which subsequent Qián-Jiā treatments of the algorithm depend.

Dating: NotBefore set at 1800; notAfter at 1810, allowing for the work’s completion in the first decade of the nineteenth century.

Translations and research

Reminder: AI generated list - beware of hallucinations. Corrections and additions welcome

  • Libbrecht, Ulrich. 1973. Chinese Mathematics in the Thirteenth Century: The Shu-shu chiu-chang of Ch’in Chiu-shao. Cambridge, Mass.: MIT Press. — Treats Zhāng Dūnrén’s recovery exposition in detail.
  • Lam Lay-Yong. 1986. “Linkages: Exploring the Similarities between the Chinese Rod Numeral System and Our Numeral System.” Archive for History of Exact Sciences 37: 365–392.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.
  • Bréard, Andrea. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs. Stuttgart: Steiner. — Treats the Jiā-Dào-era qiú-yī recovery.

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